Acrylamide is a chemical that is sometimes found in cooked starchy foods and is thought to increase the risk of certain kinds of cancer. The paper "A Statistical Regression Model for the Estimation of Acrylamide Concentrations in French Fries for Excess Lifetime Cancer Risk Assessment" describes a study to investigate the effect of frying time (in seconds) and acrylamide concentration (in micrograms per kilogram) in french fries. The data in the accompanying table are approximate values read from a graph that appeared in the paper.

\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Frying \\
Time
\end{tabular} & \begin{tabular}{c}
Acrylamide \\
Concentration
\end{tabular} \\
\hline
150 & 150 \\
\hline
240 & 115 \\
\hline
240 & 190 \\
\hline
270 & 185 \\
\hline
300 & 145 \\
\hline
300 & 270 \\
\hline
\end{tabular}

(a) Find the equation of the least-squares line for predicting acrylamide concentration using frying time. (Round your answers to four decimal places.)
[tex]$
\hat{y}=\square x+(\square x) x
$[/tex]



Answer :

To find the equation of the least-squares regression line for predicting acrylamide concentration ([tex]\(\hat{y}\)[/tex]) using frying time ([tex]\(x\)[/tex]), we need to determine the slope and intercept of the line. The equation of the line is:

[tex]\[ \hat{y} = mx + b \][/tex]

Where:
- [tex]\(m\)[/tex] is the slope of the line
- [tex]\(b\)[/tex] is the y-intercept of the line
- [tex]\(x\)[/tex] is the independent variable (frying time)
- [tex]\(\hat{y}\)[/tex] is the dependent variable (acrylamide concentration)

Given the data:
[tex]\[ \begin{array}{|c|c|} \hline \text{Frying Time (seconds)} & \text{Acrylamide Concentration (micrograms/kg)} \\ \hline 150 & 150 \\ 240 & 115 \\ 240 & 190 \\ 270 & 185 \\ 300 & 145 \\ 300 & 270 \\ \hline \end{array} \][/tex]

From the analysis, the slope ([tex]\(m\)[/tex]) and y-intercept ([tex]\(b\)[/tex]) are calculated as:
- Slope ([tex]\(m\)[/tex]): 0.4103
- Intercept ([tex]\(b\)[/tex]): 73.2692

Thus, the equation of the least-squares regression line is:

[tex]\[ \hat{y} = 0.4103x + 73.2692 \][/tex]

This equation can be used to predict the acrylamide concentration based on the frying time. Just plug in the values for [tex]\(x\)[/tex] (frying time) into the equation to find [tex]\(\hat{y}\)[/tex] (predicted acrylamide concentration).